Texture Jumps Around After Unwrap?
In the UV editor you can pin selected vertices from the UV editor menu UV -> Pin or press the P key.

You can also unpin them from the same menu or Alt + P.

There are also other ways to give finer control, for example you can create seams in your UV map effectively cutting it into parts that unwrap nicely.

Too long a discussion for here, but in mesh edit mode you can mark edges as seams, then when you unwrap it will be more forgiving.

You would still need to pin vertices if unwrapping a second time, but gives you more options.